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Figure 1: An example of the kind of question a student might receive on a standardized math assessment.

On an example like the one in Figure 1, many children select B as the “larger” angle. This relatively common misconception suggests that angle is a difficult concept for students in elementary and even middle school (Clements & Battista, 1996; Mitchelmore & White, 2000). Errors such as this one imply that when students are making judgments about the size of angles, they often use incorrect cues such as the length of an angle’s sides or the overall area contained within the angle figure (Clements & Battista, 1989; Lindquist & Kouba, 1989). Perhaps most importantly, we know that this misconception is widespread and not easily overcome despite continued instruction in school (Lehrer, Jenkins, & Osana, 1998).

So where does this kind of error come from? To find a possible explanation, we turned to the research on how children learn the meaning of words. The “whole object bias” describes a phenomenon whereby children tend to interpret a new word as referring to an unfamiliar object as a whole and not to the object’s parts or features (Markman & Hutchinson, 1984). For example, if a young child sees a rabbit for the first time, and an adult says, “Look at the rabbit!”, the child will tend to assume that “rabbit” refers to the rabbit as a whole, and not its ears or tail or the fact that it has fur. In the case of angle, this would mean that children first map the word “angle” onto the whole angle figure such that the phrase “big angle” means a big angle figure rather than an angle with a large measure of rotation.

In the current study, we hypothesized that children’s difficulty with angle does not necessarily stem from something that is inherently difficult about angles, and that word-learning assumptions that typically accelerate vocabulary growth can have surprisingly negative conceptual consequences. To address this hypothesis , we used another word-learning constraint, “mutual exclusivity”, which suggests that children are more likely to map a novel label onto a part or property of an object if they already know a separate label for the whole object (Markman, 1989). For example, after learning the word “rabbit” , children tend to assume new labels (e.g. “whiskers”) refer to a part of the rabbit. Likewise, if the whole object bias contributes to students’ difficulty with angles, providing children with a new label for the overall angle figure (e.g. “toma”), should increase the likelihood that they map “angle” onto the correct dimension of the angle figure.

Methods

Thirty 4-5-year-old children (M = 4.86 years; SD =.53) participated in a single session consisting of a pretest, training, and posttest. We chose to test 4-5-year-olds because they have not yet had any formal instruction on angles. On each pretest and posttest trial, children were presented with a card depicting two angles and asked, “Can you show me the bigger angle?” There were three trial types (Figure 2). If children initially judge angle size on the basis of any dimension aside from rotational measure, they should fail to pick the correct answer on Inconsistent trials.

Figure 2: On Consistent trials, the larger of the two angles was also formed by longer lines, on Equal trials both angles were formed by lines of the same length, and on Inconsistent trials, the larger angle was actually formed by shorter lines.

After completing the pretest, participants were randomly assigned to one of two training conditions. Importantly, participants in both conditions were given the same correct description of an angle and the same Inconsistent and Consistent practice trials with instruction and feedback. However, participants in the Experimental condition were given a separate word to refer to the whole angle figure, “toma”, and were asked to identify the “bigger toma” on a subset of the training trials. In contrast, training in the Control condition was more analogous to typical classroom training in that no separate figure label was used (Figure 3).

Figure 3: Half of the children received verbal instruction in the Control condition, and half of the children received the Experimental condition. All children saw the same stimuli.

We hypothesized that despite our explanation of the proper referent of “angle”, children in the Control condition would struggle to overcome their bias to form a whole-object interpretation of “angle.” However, children in the Experimental condition, who first learn that “toma” refers to the whole angle figure, should be more likely to search for alternate meanings of “angle”. Therefore, we predicted that children in our Experimental condition would be in a better position to understand and accept the correct meaning of “angle” that was presented during training.

Results

Our analyses aimed to address two questions. First, prior to training, do children make the type of errors commonly found in the education literature; namely do children use line length or overall figure area as a cue for judging angle size? Second, does manipulating the language used to train children on angles affect the likelihood that training is successful? Specifically, does adding a word to refer to the whole angle figure provide children with a basis to determine through mutual exclusivity that “angle” does not refer to the whole figure but rather to a particular feature of the figure?

To address the first question we looked at how children performed at pretest on the three different trial types. We found that participants scored above chance (50%) on Consistent trials and on Equal trials, but below chance on the Inconsistent trials (Figure 4). Participant’s failure on Inconsistent trials supported our hypothesis that young children tend rely on line-length or area instead of rotational measure when asked to make judgments about angle size.

Figure 4: On average, children performed significantly above chance on Consistent and Equal trial types and significantly below chance on Inconsistent trial types at pretest.

To see if our two training conditions had differential effects we looked at how children in each of the two conditions performed on the different trial types at posttest (Figure 5). We found that participants in the Experimental condition outperformed those in the Control condition on every trial type. In addition, we found that children in both conditions continued to score above chance on the Consistent trials and the Equal trials. However, on Inconsistent trials, participants in the Control condition were not significantly different from chance, while participants in the Experimental condition performed significantly above chance. Therefore, while both groups largely abandoned the strategy that resulted in below chance performance on pretest, only participants who learned a separate word for the whole angle figure consistently adopted a more successful strategy.

Figure 5: At postest, children in both conditions continued to perform above chance on the Consistent and Equal trial types, but only children in the Experimental condition performed above chance on the Inconsistent trial types.

The results suggest that even within the realm of mathematics, it can be useful to incorporate general learning principles to understand the origins of children’s pervasive misconceptions. In this particular case, evidence from the literature on word-learning helps to refute the common claim that angles are difficult for young children. Instead, our findings demonstrate that the pattern of errors we observe can be explained by the way children incorrectly interpret the word “angle”. Therefore, in practice, educators should be aware that word-learning biases, while primarily beneficial for rapid vocabulary acquisition, can have unintended conceptual implications that should be explicitly addressed in the classroom setting.